Other mistake: As Young Sheldon is a prequel series to Big Bang Theory, this episode is at odds with episode 2-13 of Big Bang Theory, "The Friendship Algorithm," in which Sheldon is unfamiliar with books like "How to Make Friends and Influence People," which he reads in this episode of Young Sheldon, produced nine years later but taking place twenty years earlier.
Factual error: The commentary suggests that Sheldon had the correct solution to problem 1 and merely lost points on the quiz because he didn't show his work. In reality he did the first problem incorrectly. Taking the derivative of (x+2) ^2+ (y+1) ^2 = 13 with respect to x yields: 2* (x+2) +2* (y+1) * (dy/dx) = 0 which implies that: dy/dx = - (x+2) / (y+1) and at the point (0,2) this means that the slope dy/dx is actually equal to - (2/3), not 2/3. It is correct that when using the form y=mx+b at the point 0,2 reveals that 2=m*0+b, or that b=2. So the correct equation for the tangent line is y = - (2/3) x + 2, not y = (2/3) x + 2 as shown in the episode. (00:13:55)
Factual error: Georgie removes his stash of cash from the air vent in his bedroom (envelope with a photo of a mustang taped to it) and the cash ($20 bills) have a serial # starting with the letter "C" which is a series of bills that were not printed until 2001. (00:15:14)
Plot hole: Near the start we see Sheldon using physics to try and take down Tam. This completely goes against all of Sheldon's phobias about touching people, especially socks. Even before this episode he has shown his displeasure at the thought of touching people. (00:05:00)
Continuity mistake: When Sheldon and Missy are sitting at the table, Missy puts a paper in the basket in front of her. A couple of shots later when she says "sounds right" the paper has disappeared. In the next shot it reappears, in a different position than it was before.
Plot hole: Memaw gets a call from Dr. Sturgis saying that he has moved to Waxahatchie and would she please call him back. He does not say the new phone number, and this was well before cell phones were in widespread use by the general public, so it could only be a land line. Memaw calls him back, although she could not know the number.